Kinematics Manipulator Kinematics P P y 1 y 1 x 1 x 1 Many - - PowerPoint PPT Presentation
Kinematics Manipulator Kinematics P P y 1 y 1 x 1 x 1 Many slides adapted from: Siegwart, Nourbakhsh and Scaramuzza, Autonomous Mobile Robots Renata Melamud, An Introduction to Robot Kinematics, CMU Rick Parent, Computer Animation,
x1 y1 P
θ
x1 y1 P
θ
Many slides adapted from: Siegwart, Nourbakhsh and Scaramuzza, Autonomous Mobile Robots Renata Melamud, An Introduction to Robot Kinematics, CMU Rick Parent, Computer Animation, Ohio State Steve Rotenberg, Computer Animation, UCSD
2
u Upcoming:
u Projects:
u Wiki permissions – http://tiny.cc/robotics-team-schedules
u Posted tonight:
u Quiz 3: Manipulation, Grasping, Kinematics u Con
cepts! s!
u Homework 2
u Resolution, Kinematics & IK, Course Progress
u Today: Inverse kinematics
3
u Forward:
u Inputs: joint angles u Outputs: coordinates of end-effector
u Inverse:
u Inputs: desired coordinates of end-effector u Outputs: joint angles
u Inverse kinematics are tricky
u Multiple solutions u No solutions u Dead spots
Joint space (robot space – previously R)
θ1, θ2, …, θn
Cartesian space (global space – previously I)
(x,y,z}, r/p/y
4
u We will sometimes use the vector Φ to represent the array of
M joint values:
u We will sometimes use the vector e to represent an array of
N joint values that describe the end effector in world space:
u Example:
u If our end effector is a full joint with orientation, e would contain 6
DOFs: 3 translations and 3 rotations. If we were only concerned with the end effector position, e would just contain the 3 translations.
M
2 1
5
joint i joint i-1 joint i+1
u Arm made up of links in a chain
u How to describe each link? u Many choices exist u DH parameters widely used
u Although it’s not true that quaternions are not widely used
u Joints each have coordinate system
u {x,y,z}, r/p/y — OR!!
u DH parameters
u Denavit-Hartenberg u ai-1, αi-1, di, θ2
6
7
u We are we looking for:
Transformation matrix T going from i to i-1:
u Determine position and orientation of end-effector
as function of displacements in joints
u Why?
u We can multiply out along all joints
i-1Ti
i i-1
8
ξI = xI yI zI θ ξR = xR yR zR θ Origin of R in I: 3 1 In 3D: 1 0 0 0 0 1 0 3 0 0 1 0 0 0 0 1 Generally: 1 0 0 x 0 1 0 y 0 0 1 z 0 0 0 1
9
ξI = x y z θI ξR = x y z θR Generally: Review? Introduction to Homogeneous Transformations & Robot Kinematics Jennifer Kay 2005
10
11
ai-1 : distance Zi-1 and Zi along Xi αi-1 : angle Zi-1 and Zi around Xi di : distance Xi-1 to Xi along Zi θ2 : angle Xi-1 and Xi around Zi
u Coordinate transformation:
screw displacement: screw displacement:
12
Transformation in DH:
13
u Goal:
u Compute the vector of joint DOFs that will cause the end
effector to reach some desired goal state
u In other words, it is the inverse previous problem
u Instead of function from world space to robot space.
1 −
ßà
14
u IK is challenging!
u f() is (usually) relatively easy to evaluate u f -1() usually isn’t
u Issues:
u There may be several possible solutions for Φ u There may be no solutions u If there is a solution, it may be expensive to find it u There are some local-minimum “stuck” configurations
u Many different approaches to solving IK problems
15
u One major way to classify IK-solving approaches:
analyt ytica ical vs numer merica ical methods
u Analytical
u Find an exact solution by directly inverting the forward
kinematics equations.
u Works on relatively simple chains.
u Numerical
u Use approximation and iteration to converge on a solution. u More expensive, more general purpose.
u We will look at one technique: Jacobians
Review adapted from: Steve Rotenberg, Computer Animation, UCSD http://graphics.ucsd.edu/courses/cse169_w05
17
u If we have a scalar function f of a single variable x, we
can write it as f(x)
u Derivative of function with respect to x is df/dx u The derivative is defined as:
x x
→ Δ → Δ
18
f-axis x-axis x f(x) Slope=df/dx
19
x x x x x x x x x x x x x x x x x dx df x x f
x x x x
2 2 lim 2 lim 2 lim lim : example For
2 2 2 2 2 2 2
= Δ + = Δ Δ + Δ = Δ − Δ + Δ + = Δ − Δ + = =
→ Δ → Δ → Δ → Δ
x
→ Δ
20
u Many algorithms require the computation of derivatives u Sometimes, we can compute them. For example: u Sometimes function is complex, can’t compute an exact derivative u As long as we can evaluate the function, we can always
approximate a derivative
2
21
f-axis x-axis Δx f(x) f(x+Δx) Slope=Δf/Δx
22
u If we know the value of a function and its derivative
at some x, we can estimate what the value of the function is at other points near x
23
u There are many mathematical and computational
approaches to finding values of x for which f(x)=0
u One such way is the gradient descent method u If we can evaluate f(x) and df/dx for any value of x,
we can always follow the gradient (slope) in the direction (currently) headed towards 0
24
u We want to find the value of x that causes f(x) to equal 0 u We will start at some value x0 and keep taking small steps:
xi+1 = xi + Δx until we find a value xN that satisfies f(xN)=0
u For each step, (try to) choose a value of Δx that gets
closer to our goal
u Use the derivative as approximation of slope of function u Use this to move ‘downhill’ towards zero
25
f-axis x-axis xi f(xi) df/dx
26
u If f(xi) is not 0, the value of f(xi) can be thought of as an error u Goal of gradient descent: minimize this error
u Making it a member of the class min
minimiza imization ion alg lgor
ithms ms
u Each step Δx results in function changing its value
u Call this Δf
u Ideally, Δf = -f(xi) – in other words, want to take a step Δx
that causes Δf to cancel out the error
u Realistically, hope each step brings us closer, and we can
eventually stop when we get close enough
u This iterative process is consistent with numerical algorithms
27
u Safety vs. efficiency
u If step size is too small, converges very slowly u If step size is too large, algorithm not reduce f.
u Because the first order approximation is valid only locally.
u If function varies widely, what is safest? u If we have a relatively smooth function? u If we feel very confident?
u We could try stepping directly to where linear
approximation passes through 0